Subgroups — C₇×#21276 #13198492

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2295	C₂	1	2	yes	maximal
3	C₇×#1969	C₂	1	2	yes	maximal
4	C₇×#1942	C₂	1	2	yes	maximal
5	C₇×#1933	C₂	1	4	yes	maximal
6	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic
7	C₄×C₇×#103	C₂	1	2	yes	maximal
8	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
11	C₄×C₇×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#103	C₂²	1	4	yes	—
15	C₂×C₇×#103	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#391	C₂²	1	4	yes	—
21	C₇×#389	C₂²	1	4	yes	—
22	C₇×#387	—	2	4	no	—
23	C₄×C₇×C₄⋊C₄	—	2	2	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₇×#376	—	2	2	no	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21276	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	C₂³	1	2	yes	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₇×Q₈	—	2	4	no	—
37	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
38	C₂×C₇×C₄⋊C₄	—	2	2	no	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²×C₇	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	D₄	1	2	yes	—
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#107	—	2	4	no	—
50	C₇×#107	C₂×C₄	1	4	yes	—
51	C₇×#103	—	2	8	no	—
52	C₇×#103	C₂×C₄	1	8	yes	—
53	C₄×C₇×C₈	—	2	4	no	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₄×C₇×C₈	—	2	2	no	—
56	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	4	no	—
60	C₇×#95	—	2	2	no	—
61	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
62	#2295	C₂×C₇	1	2	yes	—
63	#1969	C₂×C₇	1	2	yes	—
64	#1942	C₂×C₇	1	2	yes	—
65	#1933	C₂×C₇	1	4	yes	—
66	C₄×#107	C₂×C₇	1	1	yes	characteristic
67	C₄×#103	C₂×C₇	1	2	yes	—
68	C₄²×C₈	C₂×C₇	1	1	yes	characteristic
69	C₂×C₇×Q₈	—	2	2	no	—
70	C₂×C₇×Q₈	—	2	4	no	—
71	C₂×C₇×Q₈	—	2	4	no	—
72	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×Q₁₆	—	2	8	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	#41	1	4	yes	—
88	C₂×C₇×C₈	#41	1	4	yes	—
89	C₇×C₄⋊C₄	—	2	8	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	4	8	no	—
94	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
95	C₄²×C₇	—	2	8	no	—
96	C₄²×C₇	—	2	4	no	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₄²×C₇	—	4	4	no	—
101	C₄²×C₇	Q₁₆	1	4	yes	complemented
102	C₄²×C₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₇	#41	1	2	yes	—
104	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
106	C₄×Q₁₆	C₄×C₇	1	8	yes	complemented
107	C₂×#107	C₂²×C₇	1	2	yes	—
108	C₂×#107	C₂²×C₇	1	1	yes	characteristic
109	C₂×#103	C₂²×C₇	1	4	yes	—
110	C₂×#103	C₂²×C₇	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
115	#391	C₂²×C₇	1	4	yes	—
116	#389	C₂²×C₇	1	4	yes	—
117	#387	—	2	4	no	—
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
120	#376	—	2	2	no	—
121	C₄³	C₂²×C₇	1	1	yes	characteristic
122	#340	C₂²×C₇	1	2	yes	—
123	#340	C₂²×C₇	1	4	yes	—
124	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
125	C₇×Q₈	—	2	4	no	—
126	C₇×Q₈	—	2	4	no	—
127	C₇×Q₈	—	4	4	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	—	4	4	no	—
137	C₂×C₄×C₇	—	4	4	no	—
138	C₂×C₄×C₇	—	4	4	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
143	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
144	C₂×C₄×C₇	#135	1	4	yes	—
145	C₂×C₄×C₇	#135	1	2	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×#41	1	2	yes	—
148	C₂×C₄×C₇	C₂×#41	1	2	yes	—
149	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
150	C₇×C₈	—	2	2	no	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	#117	1	4	yes	—
153	C₂²×Q₈	C₂³×C₇	1	2	yes	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
156	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₇	1	2	yes	—
158	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₄×Q₈	—	2	4	no	—
160	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
164	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
168	C₂×C₄²	C₇×D₄	1	2	yes	—
169	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₇	1	2	yes	—
171	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
172	#107	—	2	4	no	—
173	#107	C₂×C₄×C₇	1	4	yes	—
174	#103	—	2	8	no	—
175	#103	C₂×C₄×C₇	1	8	yes	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
180	#95	—	2	4	no	—
181	#95	—	2	4	no	—
182	#95	—	2	4	no	—
183	#95	—	2	2	no	—
184	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₇	#532	1	2	yes	—
186	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
187	C₂²×C₇	C₂×#135	1	1	yes	characteristic
188	C₂²×C₇	C₂×#135	1	2	yes	—
189	C₄×C₇	—	2	4	no	—
190	C₄×C₇	—	2	4	no	—
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	—	4	4	no	—
193	C₄×C₇	—	4	4	no	—
194	C₄×C₇	—	8	4	no	—
195	C₄×C₇	C₄×Q₁₆	1	4	yes	—
196	C₄×C₇	C₂×#117	1	1	yes	characteristic
197	C₄×C₇	C₄×#41	1	1	yes	characteristic
198	C₄×C₇	#529	1	2	yes	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
208	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
211	Q₁₆	—	2	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
217	C₂×C₈	C₇×#41	1	4	yes	—
218	C₂×C₈	C₇×#41	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
225	C₄²	—	2	8	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₇×Q₁₆	1	4	yes	complemented
232	C₄²	C₂²×C₄×C₇	1	2	yes	—
233	C₄²	C₇×#41	1	2	yes	—
234	C₂×C₇	#2293	1	1	yes	characteristic
235	C₂×C₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
236	C₂×C₇	C₄×#135	1	1	yes	characteristic
237	C₂×C₇	#3168	1	2	yes	—
238	C₂×C₇	#3170	1	2	yes	—
239	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	2	4	no	—
242	Q₈	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
256	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
257	C₂×C₄	C₂×C₇×Q₁₆	1	4	yes	—
258	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
259	C₂×C₄	C₇×#135	1	4	yes	—
260	C₂×C₄	C₇×#135	1	2	yes	—
261	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
262	C₂×C₄	C₂×C₇×#41	1	2	yes	—
263	C₂×C₄	C₂×C₇×#41	1	2	yes	—
264	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	C₇×#117	1	4	yes	—
268	C₇	#21276	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
270	C₂²	C₇×#532	1	2	yes	—
271	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
272	C₂²	C₂×C₇×#135	1	1	yes	characteristic
273	C₂²	C₂×C₇×#135	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₄×C₇×Q₁₆	1	4	yes	—
281	C₄	C₂×C₇×#117	1	1	yes	characteristic
282	C₄	C₄×C₇×#41	1	1	yes	characteristic
283	C₄	C₇×#529	1	2	yes	—
284	C₂	C₇×#2293	1	1	yes	characteristic
285	C₂	C₂×C₄×C₇×Q₁₆	1	1	yes	characteristic
286	C₂	C₄×C₇×#135	1	1	yes	characteristic
287	C₂	C₇×#3168	1	2	yes	—
288	C₂	C₇×#3170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
11. C₄×C₇×Q₁₆	C₄	semidirect
29. #21276	C₇	direct (N × H)
101. C₄²×C₇	Q₁₆	semidirect
106. C₄×Q₁₆	C₄×C₇	semidirect
231. C₄²	C₇×Q₁₆	semidirect
268. C₇	#21276	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₇×Q₁₆	2
C₇×#2295	2
C₇×#1969	2
C₇×#1942	2
C₇×#1933	2
C₄×C₇×#107	2
C₄×C₇×#103	2
C₄²×C₇×C₈	2
#21276	7

Subgroups — C7×#21276 #13198492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21276 #13198492